Chapter 4 Applications 4.1 Arrow-Debreu Markets and Consumption Smoothing 4.1.1 The Intertemporal Budget For any given sequence {R t } t=0, pick an arbitrary q 0 > 0 and define q t recursively by q t = q 0 (1 + R 0 )(1 + R 1 )...(1 + R t ). q t represents the price of period t consumption relative to period 0 consumption. Multiplying the period-t budget by q t andaddingupoverallt, we get X q t c j t q 0 x j 0 t=0 89
George-Marios Angeletos where x j 0 (1 + R 0 )a 0 + h j 0, X h j q t 0 [w t l j t T j t ]. q 0 t=0 The above represents the intertemporal budget constraint. (1+R 0 )a j 0 is the household s financial wealth as of period 0. T j t is a lump-sum tax obligation, which may depend on the identity of household but not on its choices. h j 0 is the present value of labor income as of period 0 net of taxes; we often call h j 0 the household s human wealth as of period 0. The sum x j 0 (1 + R 0 )a j 0 + h j 0 represents the household s effective wealth. Note that the sequence of per-period budgets and the intertemporal budget constraint are equivalent. We can then write household s consumption problem as follows X max β t U(c j t,z j t ) s.t. t=0 X q t c j t q 0 x j 0 t=0 4.1.2 Arrow-Debreu versus Radner We now introduce uncertainty... Let q(s t ) be the period-0 price of a unite of the consumable in period t and event s t and w(s t ) the period-t wage rate in terms of period-t consumables for a given event s t. q(s t )w(s t ) is then the period-t and event-s t wage rate in terms of period-0 consunam- 90
Lecture Notes bles. We can then write household s consumption problem as follows max X t s.t. X t where X β t π(s t )U c j (s t ),z j (s t ) s X t s t q(s t ) c j (s t ) q 0 x j 0 h j 0 x j 0 (1 + R 0 )a 0 + h j 0, X q(s t )w(s t ) [l j (s t ) T j (s t )]. q 0 t=0 (1 + R 0 )a j 0 is the household s financial wealth as of period 0. T j (s t ) is a lump-sum tax obligation, which may depend on the identity of household but not on its choices. h j 0 is the present value of labor income as of period 0 net of taxes; we often call h j 0 the household s human wealth as of period 0. The sum x j 0 (1 + R 0 )a j 0 + h j 0 represents the household s effective wealth. 4.1.3 The Consumption Problem with CEIS Suppose for a moment that preferences are separable between consumption and leisure and are homothetic with respect to consumption: U(c, z) = u(c)+v(z). u(c) = c1 1/θ 1 1/θ Letting µ be the Lagrange multiplier for the intertemporal budget constraint, the FOCs imply β t π(s t )u 0 c j (s t ) = µq(s t ) 91
George-Marios Angeletos for all t 0. Evaluating this at t =0, we infer µ = u 0 (c j 0). It follows that q(s t ) q 0 = βt π(s t )u 0 (c j (s t )) u 0 (c j 0) µ c = β t π(s t j (s t 1/θ ) ). That is, the price of a consumable in period t relative to period 0 equals the marginal rate of intertemporal substitution between 0 and t. c j 0 Solving q t /q 0 = β t π(s t ) c j (s t )/c j 0 1/θ for c j (s t ) gives c j (s t )=c j 0 β t π(s t ) θ q(s t θ ). q 0 It follows that the present value of consumption is given by X X t s t q(s t )c j (s t )=q θ 0 cj 0 X β t π(s t ) θ q(s t ) 1 θ Substituting into the resource constraint, and solving for c 0, we conclude t=0 c j 0 = m 0 x j 0 where m 0 1 P t=0 β t π(s t ) θ [q(s t )/q 0 ] 1 θ. Consumption is thus linear in effective wealth. m 0 represent the MPC out of effective wealth as of period 0. 4.1.4 Intertemporal Consumption Smoothing, with No Uncertainty Consider for a moment the case that there is no uncertainty, so that c j (s t )=c j t and q(s t )=q t for all s t. 92
Lecture Notes Then, the riskless bond and the Arrow securities satisfy the following arbitrage condition Alternatively, q t = q 0 (1 + R 0 )(1 + R 1 )...(1 + R t ). q t = q 0 h 1+ e R 0,t i t where R e 0,t represents the average interest rate between 0 and t. Next, note that m 0 is decreasing (increasing) in q t if and only if θ>1 (θ <1). It follows that the marginal propensity to save in period 0, which is simply 1 m 0, is decreasing (increasing) in er 0,t, for any t 0, if and only if θ>1 (θ <1). A similar result applies for all t 0. We conclude Proposition 22 Suppose preferences are seperable between consumption and leisure and homothetic in consumption (CEIS). Then, the optimal consumption is linear in contemporaneous effective wealth: c j t = m t x j t where m t x j t (1 + R t )a j t + h j t, X h j q t t [w τ lτ j Tτ j ], q t τ=t 1 P τ=t βθ(τ t) (q τ /q t ) 1 θ. m t is a decreasing (increasing) function of q τ for any τ t if and only θ>1 (θ <1). That is, the marginal propensity to save out of effective wealth is increasing (decreasing) in future interest rates if and only if the elasticity of intertemporal substitution is higher (lower) 93
George-Marios Angeletos than unit. Moreover, for given prices, the optimal consumption path is independent of the timining of either labor income or taxes. Obviously, a similar result holds with uncertainty, as long as there are complete Arrow- Debreu markets. Note that any expected change in income has no effect on consumption as long as it does not affect the present value of labor income. Also, if there is an innovation (unexpected change) in income, consumption will increase today and for ever by an amount proportional to the innovation in the annuity value of labor income. To see this more clearly, suppose that the interest rate is constant and equal to the discount rate: R t = R =1/β 1 for all t. Then, the marginal propensity to consume is the consumption rule in period 0 becomes m =1 β θ (1 + R) 1 θ =1 β, c j 0 = m (1 + R)a 0 + h j 0 and the Euler condition reduces to c j t = c j 0 Therefore, the consumer choose a totally flat consumption path, no matter what is the time variation in labor income. And any unexpected change in consumption leads to a parallel shift in the path of consumption by an amount equal to the annuity value of the change in labor income. This is the manifestation of intertemporal consumption smoothing. 94
Lecture Notes More generally, if the interest rate is higher (lower) than the discount rate, the path of consumption is smooth but has a positive (negative) trend. To see this, note that the Euler condition is log c t+1 θ[β(1 + R)] θ +logc t. 4.1.5 Incomplete Markets and Self-Insurance The above analysis has assumed no uncertainty, or that markets are complete. Extending the model to introduce idiosyncratic uncertainty in labor income would imply an Euler condition of the form u 0 (c j t)=β(1 + R)E t u 0 (c j t+1) Note that, because of the convexity of u 0, as long as Var t [c j t+1] > 0, we have E t u 0 (c j t+1) > u 0 (E t c j t+1) and therefore E t c j t+1 > [β(1 + R)] θ c j t This extra kick in consumption growth reflects the precautionary motive for savings. It remains true that transitory innovations in income result to persistent changes in consumption (because of consumption smoothing). At the same time, consumers find it optimal to accumulate a buffer stock, as a vehicle for self-insurance. 4.2 Aggregation and the Representative Consumer Consider a deterministic economy populated by many heterogeneous households. Households differ in their initial asset positions and (perhaps) their streams of labor income, but not in their preferences. They all have CEIS preferences, with identical θ. 95
George-Marios Angeletos Following the analysis of the previous section, consumption for individual j is given by c j t = m t x j t. Note that individuals share the same MPC out of effective wealth because they have identical θ. Adding up across households, we infer that aggregate consumption is given by c t = m t x t where m t x t (1 + R t )a t + h t, X q τ h t [w τ l τ T τ ], q t τ=t 1 P τ=t βθ(τ t) (q τ /q t ) 1 θ. Next, recall that individual consumption growth satisfies q t = βt u 0 (c j t) q 0 u 0 (c j 0) Ã! 1/θ = β t c j t c j, 0 for every j. But if all agents share the same consumption growth rate, this should be the aggregate one. Therefore, equilibrium prices and aggregate consumption growth satisfy Equivalently, q t q 0 = β t µ 1/θ ct c 0 q t = βt u 0 (c t ) q 0 u 0 (c 0 ). 96
Lecture Notes Consider now an economy that has a single consumer, who is endowed with wealth x t and has preferences U(c) = The Euler condition for this consumer will be c1 1/θ 1 1/θ q t = βt u 0 (c t ) q 0 u 0 (c 0 ). Moreover, this consumer will find it optimal to choose consumption c t = m t x t. But these are exactly the aggregative conditions we found in the economy with many agents. That is, the two economies share exactly the same equilibrium prices and allocations. It is in this sense that we can think of the single agent of the second economy as the representative agent of the first multi-agent economy. Note that here we got a stronger result than just the existence of a representative agent. Not only a representative agent existed, but he also had exactly the same preferences as each of the agents of the economy. This was true only because agents had identical preference to start with and their preferences were homothetic. If either condition fails, the preferences of the representative agent will be weighted average of the population preferences, with the weights depending on the wealth distribution. Finally, note that these aggregation results extend easily to the case of uncertainty as long as markets are complete. 97
4.3 Fiscal Policy 4.3.1 Ricardian Equilivalence George-Marios Angeletos The intertemporal budget for the representative household is given by where and a 0 = k 0 + b 0. X q t c t q 0 x 0 t=0 x 0 =(1+R 0 )a 0 + X t=0 q t q 0 [w t l t T t ] On the other hand, the intertemporal budget constraint for the government is X q t g t + q 0 (1 + R 0 )b 0 = t=0 X q t T t t=0 Substituting the above into the formula for x 0, we infer x 0 =(1+R 0 )k 0 + X t=0 q t q 0 w t l t X t=0 q t q 0 g t That is, aggregate household wealth is independent of either the outstanding level of public debt or the timing of taxes. We can thus rewrite the representative household s intertemporal budget as X q t [c t + g t ] q 0 (1 + R 0 )k 0 + t=0 X q t w t l t t=0 Since the representative agent s budget constraint is independent of either b 0 or {T t } t=0, his consumption and labor supply will also be independent. But then the resource constraint implies that aggregate investment will be unaffected as well. Therefore, the 98
Lecture Notes aggregate path {c t,k t } t=0 is independent of either b 0 or {T t } t=0. All that matter is the stream of government spending, not the way this is financed. More generally, consider now arbitrary preferences and endogenous labor supply, but suppose that the tax burden and public debt is uniformly distributed across households. Then, for every individual j, effective wealth is independent of either the level of public debt or the timing of taxes: x j 0 =(1+R 0 )k j 0 + X t=0 q t q 0 w t l j t X t=0 q t q 0 g t, Since the individual s intertemporal budget is independent of either b 0 or {T t } t=0, her optimal plan {c j t,lt,a j j t} t=0 will also be independent of either b 0 or {T t } t=0 for any given price path. But if individual behavior does not change for given prices, markets will continue to clear for the same prices. That is, equilibrium prices are indeed also independent of either b 0 or {T t } t=0. We conclude Proposition 23 Equilibrium prices and allocations are independent of either the intial level of public debt, or the mixture of deficits and (lump-sum) taxes that the government uses to finance governement spending. Remark: For Ricardian equivalence to hold, it is critical both that markets are complete (so that agents can freely trade the riskless bond) and that horizons are infinite (so that the present value of taxes the household expects to pay just equals the amount of public debt it holds). If either condition fails, such as in OLG economies or economies with borrowing constraints, Ricardian equivalence will also fail. Ricardian equivalence may also fail if there are 99
George-Marios Angeletos 4.3.2 Tax Smoothing and Debt Management topic covered in class notes to be completed 4.4 Risk Sharing and CCAPM 4.4.1 Risk Sharing topic covered in class notes to be completed 4.4.2 Asset Pricing and CCAPM topic covered in class notes to be completed 4.5 Ramsey Meets Tobin: Adjustment Costs and q topic covered in recitation notes to be completed 4.6 Ramsey Meets Laibson: Hyperbolic Discounting 4.6.1 Implications for Long-Run Savings topic covered in class 100
Lecture Notes notes to be completed 4.6.2 Implications for Self-Insurance topic covered in class notes to be completed 101
George-Marios Angeletos 102
Chapter 5 Overlapping Generations Models 5.1 OLG and Life-Cycle Savings 5.1.1 Households Consider a household born in period t, living in periods t and t +1. We denote by c y t his consumption when young and c o t+1 his consumption when old. Preferences are given by u(c y t )+βu(c o t+1 ) where β denotes a discount factor and u is a neoclassical utility function. The household is born with zero initial wealth, saves only for life-cycle consumption smoothing, and dies leaving no bequests to future generations. The household receives labor income possibly in both periods of life. We denote by l y and l o the endowments of effective labor when young and when old, respectively. The budget constraint during 103
George-Marios Angeletos the first period of life is thus c y t + a t w t l y, whereas the budget constraint during the second period of life is c o t+1 w t+1l o +(1+R t+1 )a t. Adding up the two constraints (and assuming that the household can freely borrow and lend when young, so that a t can be either negative or positive), we derive the intertemporal budget constraint of the household: c y t + co t+1 1+R t+1 h t w t l y + w t+1l o 1+R t+1 The household choose consumption and savings so as to maximize life utility subject to his intertemporal budget: max u(c y t )+βu(c o t+1 ) The Euler condition gives: s.t. c y t + co t+1 1+R t+1 h t. u 0 (c y t )=β(1 + r t+1 )u 0 (c o t+1). In words, the household chooses savings so as to smooth (the marginal utility of) consumption over his life-cycle. With CEIS preferences, u(c) =c 1 1/θ /(1 1/θ), the Euler condition reduces to c o t+1 c y t =[β(1 + R t+1 )] θ. 104
Lecture Notes Life-cycle consumption growth is thus an increasing function of the return on savings and the discount factor. Combining with the intertemporal budget, we infer h t = c y t + co t+1 1+R t+1 = c y t + β θ (1 + R t+1 ) θ 1 c y t and therefore optimal consumption during youth is given by c y t = m(r t+1 ) h t where 1 m(r) 1+β θ (1 + R). θ 1 Finally, using the period-1 budget, we infer that optimal life-cycle saving are given by a t = w t l y m(r t+1 )h t =[1 m(r t+1 )]w t l y m(r t+1 ) w t+1l o 1+R t+1 5.1.2 Population Growth We denote by N t the size of generation t and assume that population grows at constant rate n : N t+1 =(1+n)N t It follows that the size of the labor force in period t is L t = N t l y + N t 1 l o = N t l y + lo 1+n We henceforth normalize l y + l o /(1 + n) =1, so that L t = N t. Remark: As always, we can reinterpret N t as effective labor and n as the growth rate of exogenous technological change. 105
George-Marios Angeletos 5.1.3 Firms and Market Clearing Let k t = K t /L t = K t /N t. The FOCs for competitive firms imply: r t = f 0 (k t ) r(k t ) w t = f(k t ) f 0 (k t )k t w(k t ) On the other hand, the arbitrage condition between capital and bonds implies 1+R t = 1+r t δ, and therefore R t = f 0 (k t ) δ r(k t ) δ Total capital is given by the total supply of savings: K t+1 = a t N t Equivalently, (1 + n)k t+1 = a t. 5.1.4 General Equilibrium Combining (1 + n)k t+1 = a t with the optimal rule for savings, and substituting r t = r(k t ) and w t = w(k t ), we infer the following general-equilibrium relation between savings and capital in the economy: (1 + n)k t+1 =[1 m(r(k t+1 ) δ)]w(k t )l y w(k t+1 )l o m(r(k t+1 ) δ) 1+r(k t+1 ) δ. We rewrite this as an implicit relation between k t+1 and k t : Φ(k t+1,k t )=0. 106
Lecture Notes Note that Φ 1 = (1+n)+h m r R k + µ w mlo, k 1+r Φ 2 = (1 m) w k ly. Recall that m R r 0 θ 1, whereas k = F KK < 0, w k = F LK > 0, and w k 1+r > 0. It follows that Φ 2 is necessarily negative, but Φ 1 may be of either sign: Φ 2 < 0 but Φ 1 0. We can thus always write k t as a function of k t+1, but to write k t+1 as a function of k t,we need Φ to be monotonic in k t+1. Asufficient condition for the latter to be the case is that savings are non-decreasing in real returns: θ 1 m r 0 Φ 1 > 0 In that case, we can indeed express k t+1 as a function of k t : k t+1 = G(k t ). Moreover, G 0 = Φ 2 Φ 1 > 0, and therefore k t+1 increases monotonically with k t. However, there is no guarantee that G 0 < 1. Therefore, in general there can be multiple steady states (and poverty traps). See Figure 1. On the other hand, if θ is sufficiently lower than 1, the equation Φ(k t+1,k t )=0may have multiple solutions in k t+1 for given k t. That is, it is possible to get equilibrium indeterminacy. Multiple equilibria indeed take the form of self-fulfilling prophesies. The anticipation of a high capital stock in the future leads agents to expect a low 107
George-Marios Angeletos return on savings, which in turn motivates high savings (since θ<1) and results to a high capital stock in the future. Similarly, the expectation of low k in period t +1 leads to high returns and low savings in the period t, which again vindicates initial expectations. See Figure 2. 5.2 Some Examples 5.2.1 Log Utility and Cobb-Douglas Technology Assume that the elasticity of intertemporal substitution is unit, that the production technology is Cobb-Douglas, and that capital fully depreciates over the length of a generation: It follows that the MPC is constant, u(c) =lnc, f(k) =k α, and δ =1. m = 1 1+β and one plus the interest rate equals the marginal product of capital, where 1+R =1+r(k) δ = r(k) r(k) = f 0 (k) =αk α 1 w(k) = f(k) f 0 (k)k =(1 α)k α. Substituting into the formula for G, we conclude that the law of motion for capital reduces to k t+1 = G(k t )= f 0 (k t )k t ζ(1 + n) = αkα t ζ(1 + n) 108
Lecture Notes where the scalar ζ>0 is given by ζ (1 + β)α +(1 α)lo /(1 + n) β(1 α)l y Note that ζ is increasing in l o,decreasinginl y, decreasing in β, and increasing in α (decreasing in 1 α). Therefore, G (savings) decreases with an increase in l o and a decrease in l y, with an decrease in β, or with an increase in α. 5.2.2 Steady State The steady state is any fixed point of the G mapping: k olg = G(k olg ) Using the formula for G, we infer f 0 (k olg )=ζ(1 + n) and thus k olg =(f 0 ) 1 (ζ(1 + n)). Recall that the golden rule is given by f 0 (k gold )=δ + n, and here δ =1. That is, k gold =(f 0 ) 1 (1 + n). Pareto optimality requires k olg <k gold r>δ+ n ζ>1, while Dynamic Inefficiency occurs when k olg >k gold r<δ+ n ζ<1. 109
George-Marios Angeletos Note that ζ = (1 + β)α +(1 α)lo /(1 + n) β(1 α)l y is increasing in l o,decreasinginl y, decreasing in β, and increasing in α (decreasing in 1 α). Therefore, inefficiency is less likely the higher l o, the lower l y, the lower is β, and the higher α. Provide intuition... In general, ζ canbeeitherhigherorlowerthan1. Thereisthusnoguaranteethat there will be no dynamic inefficiency. But, Abel et al argue that the empirical evidence suggests r>δ+ n, and therefore no evidence of dynamic inefficiency. 5.2.3 No Labor Income When Old: The Diamond Model Assume l o =0and therefore l y =1. That is, household work only when young. This case corresponds to Diamond s OLG model. In this case, ζ reduces to (1 + β)α ζ = β(1 α). ζ is increasing in α and ζ =1 α = 1. Therefore, 2+1/β r n + δ ζ 1 α (2 + 1/β) 1 Note that, if β (0, 1), then (2+1/β) 1 (0, 1/3) and therefore dynamic inefficiency is possible only if α is sufficiently lower than 1/3. This suggests that dynamic inefficiency is rather unlikely. However, in an OLG model β can be higher than 1, and the higher β the more likely to get dynamic inefficiency in the Diamond model. 110
Lecture Notes Finally, note that dynamic inefficiency becomes less likely as we increase l o, that is, as we increase income when old (hint: retirement benefits). 5.2.4 Perpetual Youth: The Blanchard Model We now reinterpret n as the rate of exogenous technological growth. We assume that household work the same amount of time in every period, meaning that in effective terms l o = (1 + n)l y. Under the normalization l y + l o /(1 + n) =1,weinferl y = l o /(1 + n) =1/2. The scalar ζ reduces to ζ = 2(1 + β)α +(1 α) β(1 α) Note that ζ is increasing in α, andsinceα>0, we have ζ> 2(1 + β)0 + (1 0) β(1 0) = 1 β. If β (0, 1), it is necessarily the case that ζ>1. It follows that necessarily r>n+ δ and thus k blanchard <k gold, meaning that it is impossible to get dynamic inefficiency. Moreover, recall that the steady state in the Ramsey model is given by β[1 + f 0 (k ramsey ) δ] =1+n f 0 (k ramsey )=(1+n)/β k ramsey =(f 0 ) 1 ((1 + n)/β) 111
George-Marios Angeletos while the OLG model has f 0 (k blanchard )=ζ(1 + n) k blanchard =(f 0 ) 1 (ζ(1 + n)) Since ζ>1/β, we conclude that the steady state in Blanchard s model is necessarily lower than in the Ramsey model. We conclude k blanchard <k ramsey <k gold. Discuss the role of perpetual youth and new-comers. 5.3 Ramsey Meets Diamond: The Blanchard Model topic covered in recitation notes to be completed 5.4 Various Implications Dynamic inefficiency and the role of government Ricardian equivalence breaks, public debt crowds out investment. Fully-funded social security versus pay-as-you-go. Bubbles notes to be completed 112
Chapter 6 Endogenous Growth I: AK, H, and G 6.1 The Simple AK Model 6.1.1 Pareto Allocations Total output in the economy is given by Y t = F (K t,l t )=AK t, where A>0 is an exogenous parameter. In intensive form, y t = f(k t )=Ak t. The social planner s problem is the same as in the Ramsey model, except for the fact that output is linear in capital: X max u(c t ) t=0 s.t. c t + k t+1 f(k t )+(1 δ)k t 113
George-Marios Angeletos The Euler condition gives u 0 (c t ) = β (1 + A δ) u 0 (c t+1 ) Assuming CEIS, this reduces to c t+1 c t =[β (1 + A δ)] θ or ln c t+1 ln c t = θ(r ρ) where R = A δ is the net social return on capital. That is, consumption growth is proportional to the difference between the real return on capital and the discount rate. Note that now the real return is a constant, rather than diminishing with capital accumulation. Note that the resource constraint can be rewritten as c t + k t+1 =(1+A δ)k t. Since total resources (the RHS) are linear in k, an educated guess is that optimal consumption and investment are also linear in k. We thus propose c t =(1 s)(1 + A δ)k t k t+1 = s(1 + A δ)k t where the coefficient s is to be determined and must satisfy s (0, 1) for the solution to exist. It follows that c t+1 c t = k t+1 k t = y t+1 y t 114
Lecture Notes so that consumption, capital and income all grow at the same rate. To ensure perpetual growth, we thus need to impose β (1 + A δ) > 1, or equivalently A δ>ρ.if that condition were not satisfied, and instead A δ<ρ, then the economy would shrink at a constant rate towards zero. Fromtheresourceconstraintwethenhave c t k t + k t+1 k t =(1+A δ), implying that the consumption-capital ratio is given by c t k t =(1+A δ) [β (1 + A δ)] θ Using c t =(1 s)(1 + A δ)k t and solving for s we conclude that the optimal saving rate is s = β θ (1 + A δ) θ 1. Equivalently, s = β θ (1 + R) θ 1, where R = A δ is the net social return on capital. Note that the saving rate is increasing (decreasing) in the real return if and only if the EIS is higher (lower) than unit, and s = β for θ =1. Finally, to ensure s (0, 1), we impose β θ (1 + A δ) θ 1 < 1. This is automatically ensured when θ 1 and β (1 + A δ) > 1, as then s = β θ (1 + A δ) θ 1 β < 1. But when θ > 1, this puts an upper bound on A. If A exceeded that upper bound, then the social planner could attain infinite utility, and the problem is not well-defined. 115
George-Marios Angeletos We conclude that Proposition 24 Consider the social planner s problem with linear technology f(k) =Ak and CEIS preferences. Suppose (β,θ,a,δ) satisfy β (1 + A δ) > 1 >β θ (1 + A δ) θ 1. Then, the economy exhibits a balanced growth path. Capital, output, and consumption all grow at a constant rate given by k t+1 k t = y t+1 y t = c t+1 c t =[β (1 + A δ)] θ > 1. while the investment rate out of total resources is given by s = β θ (1 + A δ) θ 1. The growth rate is increasing in the net return to capital, increasing in the elasticity of intertemporal substitution, and decreasing in the discount rate. 6.1.2 The Frictionless Competitive Economy Consider now how the social planner s allocation is decentralized in a competitive market economy. Suppose that the same technology that is available to the social planner is available to each single firm in the economy. Then, the equilibrium rental rate of capital and the equilibrium wage rate will be given simply r = A and w =0. The arbitrage condition between bonds and capital will imply that the interest rate is R = r δ = A δ. 116
Lecture Notes Finally, the Euler condition for the household will give c t+1 c t =[β (1 + R)] θ. We conclude that the competitive market allocations coincide with the Pareto optimal plan. Note that this is true only because the private and the social return to capital coincide. 6.2 A Simple Model of Human Capital 6.2.1 Pareto Allocations Total output in the economy is given by Y t = F (K t,h t )=F(K t,h t L t ), where F is a neoclassical production function, K t is aggregate capital in period t, h t is human capital per worker, and H t = h t L t is effective labor. Note that, due to CRS, we can rewrite output per capita as µ kt ht y t = F (k t,h t )=F, 1 [k t + h t ]= h t k t + h t or equivalently y t = F (k t,h t )=A (κ t )[k t + h t ], where κ t = k t /h t = K t /H t is the ratio of physical to human capital, k t + h t measures total capital, and A (κ) represents the return to total capital. F (κ, 1) 1+κ f(κ) 1+κ 117
George-Marios Angeletos Total output can be used for consumption or investment in either type of capital, so that the resource constraint of the economy is given by c t + i k t + ih t y t. The laws of motion for two types of capital are k t+1 = (1 δ k )k t + i k t h t+1 = (1 δ h )h t + i h t As long as neither i k t nor i h t are constrained to be positive, the resource constraint and the two laws of motion are equivalent to a single constraint, namely c t + k t+1 + h t+1 F (k t,h t )+(1 δ k )k t +(1 δ h )h t The social planner s problem thus becomes X max u(c t ) t=0 s.t. c t + k t+1 + h t+1 F (k t,h t )+(1 δ k )k t +(1 δ h )h t Since there are two types of capital, we have two Euler conditions, one for each type of capital. The one for physical capital is while the one for human capital is u 0 (c t ) u 0 (c t+1 ) = β [1 + F k(k t+1,h t+1 ) δ k ], u 0 (c t ) u 0 (c t+1 ) = β [1 + F h(k t+1,h t+1 ) δ h ]. 118
Lecture Notes Combining the two Euler condition, we infer F k (k t+1,h t+1 ) δ k = F h (k t+1,h t+1 ) δ h. Remember that F is CRS, implying that both F k and F h are functions of the ratio κ t+1 = k t+1 /h t+1. In particular, F k is decreasing in κ and F h is increasing in κ. The above condition therefore determines a unique optimal ratio κ such that k t+1 h t+1 = κ t+1 = κ for all t 0. For example, if F (k, h) =k α h 1 α and δ k = δ h, then F k F h = α h 1 α k and therefore κ = α. More generally, the optimal physical-to-human capital ratio is 1 α increasing in the relative productivity of physical capital and decreasing in the relative depreciation rate of physical capital. Multiplying the Euler condition for k with k t+1 /(k t+1 + h t+1 ) and the one for h with h t+1 /(k t+1 + h t+1 ), and summing the two together, we infer the following weighted Euler condition: u 0 (c t ) u 0 (c t+1 ) = β By CRS, we have ½ 1+ k t+1[f k (k t+1,h t+1 ) δ k ]+h t+1 [F h (k t+1,h t+1 ) δ h ] k t+1 + h t+1 ¾ F k (k t+1,h t+1 )k t+1 + F h (k t+1,h t+1 )h t+1 = F (k t+1,h t+1 )=A (κ t+1 )[k t+1 + h t+1 ] It follows that ½ u 0 (c t ) u 0 (c t+1 ) = β 1+A (κ t+1 ) δ ¾ kk t+1 + δ h h t+1 k t+1 + h t+1 Using the fact that κ t+1 = κ, and letting A A (κ ) F (κ, 1) 1+κ 119
George-Marios Angeletos represent the effective return to total capital and δ κ 1+κ δ k + 1 1+κ δ h the effective depreciation rate of total capital, we conclude that the weighted Euler condition evaluated at the optimal physical-to-human capital ratio is u 0 (c t ) u 0 (c t+1 ) = β [1 + A δ ]. Assuming CEIS, this reduces to c t+1 c t =[β (1 + A δ )] θ or ln c t+1 ln c t = θ(a δ ρ) wherea δ is the net social return to total savings. Note that the return is constant along the balanced growth path, but it is not exogenous. It instead depends on the ratio of physical to human capital. The latter is determined optimally so as to maximize the net return on total savings. To see this, note that k t+1 /h t+1 = κ indeed solves the following problem max F (k t+1,h t+1 ) δ k k t+1 δ h h t+1 s.t. k t+1 + h t+1 = constant Given the optimal ratio κ, the resource constraint can be rewritten as c t +[k t+1 + h t+1 ]=(1+A δ )[k t + h t ]. 120
Lecture Notes Like in the simple Ak model, an educated guess is then that optimal consumption and total investment are also linear in total capital: c t =(1 s)(1 + A δ )[k t + h t ], k t+1 + h t+1 = s(1 + A δ )[k t + h t ]. The optimal saving rate s is given by s = β θ (1 + A δ ) θ 1. We conclude that Proposition 25 Consider the social planner s problem with CRS technology F (k, h) over physical and human capital and CEIS preferences. Let κ be the ratio k/h that maximizes F (k, h) δ k k δ h h for any given k + h, and let A F (κ, 1) 1+κ and δ κ 1+κ δ k + 1 1+κ δ h Suppose (β,θ,f,δ k,δ h ) satisfy β (1 + A δ ) > 1 >β θ (1 + A δ ) θ 1. Then, the economy exhibits a balanced growth path. Physical capital, human capital, output, and consumption all grow at a constant rate given by y t+1 y t = c t+1 c t =[β (1 + A δ )] θ > 1. while the investment rate out of total resources is given by s = β θ (1 + A δ ) θ 1 and the optimal ratio of physical to human capital is k t+1 /h t+1 = κ. Thegrowthrateisincreasing in the productivity of either type of capital, increasing in the elasticity of intertemporal substitution, and decreasing in the discount rate. 121
George-Marios Angeletos 6.2.2 Market Allocations Consider now how the social planner s allocation is decentralized in a competitive market economy. The household budget is given by c t + i k t + ih t + b t+1 y t +(1+R t )b t. and the laws of motion for the two types of capital are k t+1 = (1 δ k )k t + i k t h t+1 = (1 δ h )h t + i h t We can thus write the household budget as c t + k t+1 + h t+1 + b t+1 (1 + r t δ k )k t +(1+w t δ h )h t +(1+R t )b t. Note that r t δ k and w t δ h represent the market returns to physical and human capital, respectively. Suppose that the same technology that is available to the social planner is available to each single firm in the economy. Then, the equilibrium rental rate of capital and the equilibrium wage rate will be given simply r t = F k (κ t, 1) and w t = F h (κ t, 1), where κ t = k t /h t. The arbitrage condition between bonds and the two types of capital imply that R t = r t δ k = w t δ h. 122
Lecture Notes Combining the above with the firms FOC, we infer F k (κ t, 1) F h (κ t, 1) = r t w t = δ h δ k and therefore κ t = κ, like in the Pareto optimum. It follows then that R t = A δ, where A and δ are defined as above. Finally, the Euler condition for the household is Using R t = A δ, we conclude u 0 (c t ) u 0 (c t+1 ) = β (1 + R t). y t+1 y t = c t+1 c t =[β (1 + A δ )] θ Hence, the competitive market allocations once again coincide with the Pareto optimal plan. Note that again this is true only because the private and the social return to each type of capital coincide. 6.3 Learning by Education (Ozawa and Lucas) see problem set notes to be completed 123
George-Marios Angeletos 6.4 Learning by Doing and Knowledge Spillovers (Arrow and Romer) 6.4.1 Market Allocations Output for firm m is given by Y m t = F (K m t,h tl m t ) where h t represents the aggregate level of human capital or knowledge. h t is endogenously determined in the economy (we will specify in a moment how), but it is taken as exogenous from either firms or households. Firm profits are given by Π m t = F (Kt m,h tl m t ) r tkt m w t L m t The FOCs give r t = F K (Kt m,h tl m t ) w t = F L (Kt m,h tl m t ) h t Using the market clearing conditions for physical capital and labor, we infer Kt m /L m t = k t, where k t is the aggregate capital labor ratio in the economy. We conclude that, given k t and h t, market prices are given by r t = F K (k t,h t )=f 0 (κ t ) w t = F L (k t,h t )h t =[f(κ t ) f 0 (κ t )κ t ]h t where f(κ) F (κ, 1) is the production function in intensive form and κ t = k t /h t. 124
Lecture Notes Households, like firms, take w t,r t and h t as exogenously given. household maximizes utility subject to the budget constraint The representative c t + k t+1 + b t+1 w t +(1+r t δ)k t +(1+R t )b t. Arbitrage between bonds and capital imply R t = r t δ and the Euler condition gives u 0 (c t ) u 0 (c t+1 ) = β (1 + R t)=β(1 + r t δ). To close the model, we need to specify how h t is determined. Following Arrow and Romer, we assume that knowledge accumulation is the unintentional by-product of learning-by-doing in production. We thus let the level of knowledge to be proportional to either the level of output, or the level of capital: for some constant η>0. h t = ηk t, It follows that the ratio k t /h t = κ t is pinned down by κ t =1/η. Letting the constants A and ω be defined A f 0 (1/η) and ω f(1/η)η f 0 (1/η), we infer that equilibrium prices are given by Substituting into the Euler condition gives r t = A and w t = ωk t. u 0 (c t ) = β (1 + A δ). u 0 (c t+1 ) Finally, it is immediate that capital and output grow at the same rate as consumption. We conclude 125
George-Marios Angeletos Proposition 26 Let A f 0 (1/η) and ω f(1/η)η f 0 (1/η), and suppose β (1 + A δ) > 1 >β θ (1 + A δ) θ 1. Then, the market economy exhibits a balanced growth path. Physical capital, knowledge, output, wages, and consumption all grow at a constant rate given by y t+1 y t = c t+1 c t =[β (1 + A δ)] θ > 1. The wage rate is given by w t = ωk t, while the investment rate out of total resources is given by s = β θ (1 + A δ) θ 1. 6.4.2 Pareto Allocations and Policy Implications Consider now the Pareto optimal allocations. The social planner recognizes that knowledge in the economy is proportional to physical capital and internalizes the effect of learning-by-doing. He thus understands that output is given by y t = F (k t,h t )=A k t where A f(1/η)η = A + ω represents the social return on capital. It is therefore as if the social planner had access to a linear technology like in the simple Ak model, and therefore the Euler condition for the social planner is given by u 0 (c t ) u 0 (c t+1 ) = β (1 + A δ). Note that the social return to capital is higher than the private (market) return to capital: A >A= r t The difference is actually ω, the fraction of the social return on savings that is wasted as labor income. 126
Lecture Notes Proposition 27 Let A A+ω f(1/η)η and suppose β (1 + A δ) > 1 >β θ (1 + A δ) θ 1. Then, the Pareto optimal plan exhibits a balanced growth path. Physical capital, knowledge, output, wages, and consumption all grow at a constant rate given by y t+1 y t = c t+1 c t =[β (1 + A δ)] θ > 1. Note that A<A, and therefore the market growth rate is lower than the Pareto optimal one. Exercise: Reconsider the market allocation and suppose the government intervenes by subsidizing either private savings or firm investment. Find, in each case, what is the subsidy that implements the optimal growth rate. Is this subsidy the optimal one, in the sense that it maximizes social welfare? 6.5 Government Services (Barro) notes to be completed 127
George-Marios Angeletos 128
Chapter 7 Endogenous Growth II: R&D and Technological Change 7.1 Expanding Product Variety: The Romer Model There are three production sectors in the economy: A final-good sector, an intermediate good sector, and an R&D sector. The final good sector is perfectly competitive and thus makes zero profits. Its output is used either for consumption or as input in each of the other two sector. The intermediate good sector is monopolistic. There is product differentiation. Each intermediate producer is a quasi-monopolist with respect to his own product and thus enjoys positive profits. To become an intermediate producer, however, you must first acquire a blueprint from the R&D sector. A blueprint is simply the technology or know-how for transforming final goods to differentiated intermediate inputs. 129
George-Marios Angeletos The R&D sector is competitive. Researchers produce blueprints, that is, the technology for producing an new variety of differentiated intermediate goods. Blueprints are protected by perpetual patents. Innovators auction their blueprints to a large number of potential buyers, thus absorbing all the profits of the intermediate good sector. But there is free entry in the R&D sector, which drive net profits in that sector to zero as well. 7.1.1 Technology The technology for final goods is given by a neoclassical production function of labor L and a composite factor Z: Y t = F (X t,l t )=A(L t ) 1 α (X t ) α. The composite factor is given by a CES aggregator of intermediate inputs: X t = Z Nt 0 (X t,j ) ε dj 1/ε, where N t denotes the number of different intermediate goods available in period t and X t,j denotes the quantity of intermediate input j employed in period t. In what follows, we will assume ε = α, which implies Z Nt Y t = A(L t ) 1 α (X t,j ) α dj Note that ε = α means that the elasticity of substitution between intermediate inputs is 1 and therefore the marginal product of each intermediate input is independent of the quantity of other intermediate inputs: Y t X t,j = αa 130 0 µ Lt X t,j 1 α.
Lecture Notes More generally, intermediate inputs could be either complements or substitutes, in the sense that the marginal product of input j could depend either positively or negatively on X t. We will interpret intermediate inputs as capital goods and therefore let aggregate capital be given by the aggregate quantity of intermediate inputs: K t = Z Nt 0 X t,j dj. Finally, note that if X t,j = X for all j and t, then Y t = AL 1 α t N t X α and implying K t = N t X, Y t = A(N t L t ) 1 α (K t ) α or, in intensive form, y t = ANt 1 α kt α. Therefore, to the extent that all intermediate inputs are used in the same quantity, the technology is linear in knowledge N and capital K. Therefore, if both N and K grow at a constant rate, as we will show to be the case in equilibrium, the economy will exhibit long run growth like in an Ak model. 7.1.2 Final Good Sector The final good sector is perfectly competitive. Firms are price takers. Final good firms solve max Y t w t L t 131 Z Nt 0 (p t,j X t,j )dj
George-Marios Angeletos where w t is the wage rate and p t,j is the price of intermediate good j. Profits in the final good sector are zero, due to CRS, and the demands for each input aregivenbythefocs and for all j [0,N t ]. p t,j = w t = Y t L t =(1 α) Y t L t Y µ 1 α t Lt = αa X t,j X t,j 7.1.3 Intermediate Good Sector The intermediate good sector is monopolistic. downward sloping demand for their output. Firms understand that they face a The producer of intermediate good j solves subject to the demand curve max Π t,j = p t,j X t,j κ(x t,j ) µ 1 αa 1 α X t,j = L t, p t,j where κ(x) represents the cost of producing X in terms of final-good units. We will let the cost function be linear: κ(x) =X. The implicit assumption behind this linear specification is that technology of producing intermediate goods is identical to the technology of producing final goods. Equivalently, 132
Lecture Notes you can think of intermediate good producers buying final goods and transforming them to intermediate inputs. What gives them the know-how for this transformation is precisely the blueprint they hold. The FOCs give for the optimal price, and for the optimal supply, where p t,j = p 1 α > 1 X t,j = xl x A 1 1 α α 2 1 α. Note that the price is higher than the marginal cost (p =1/α > κ 0 (X) =1),the gap representing the mark-up that intermediate-good firms charge to their customers (the final good firms). Because there are no distortions in the economy other than monopolistic competition in the intermediate-good sector, the price that final-good firms are willing to pay represents the social product of that intermediate input and the cost that intermediate-good firms face represents the social cost of that intermediate input. Therefore, the mark-up 1/α gives the gap between the social product and the social cost of intermediate inputs. (Hint: The social planner would like to correct for this distortion. How?) The resulting maximal profits are Π t,j = πl where π (p 1)x = 1 α α x = 1 α α A 1 2 1 α α 1 α. 133
George-Marios Angeletos 7.1.4 The Innovation Sector The present value of profits of intermediate good j from period t andonisgivenby V t,j = X τ=t q τ q t Π τ,j or recursively V t,j = Π t,j + V t+1,j 1+R t+1 We know that profits are stationary and identical across all intermediate goods: Π t,j = πl for all t, j. As long as the economy follows a balanced growth path, we expect the interest rate to be stationary as well: R t = R for all t. It follows that the present value of profits is stationary and identical across all intermediate goods: V t,j = V = πl R. Equivalently, RV = πl, which has a simple interpretation: The opportunity cost of holding an asset which has value V and happens to be a blueprint, instead of investing in bonds, is RV ; the dividend that this asset pays in each period is πl; arbitrage then requires the dividend to equal the opportunity cost of the asset, namely RV = πl. New blueprints are also produced using the same technology as final goods. In effect, innovators buy final goods and transform them to blueprints at a rate 1/η. Producing an amount N of new blueprints costs η N, where η>0measures the cost of R&D in units of output. On the other hand, the value of these new blueprints is V N, where V = πl/r. Net profits for a research firm are thus given by (V η) N 134
Lecture Notes Free entry in the sector of producing blueprints then implies V = η. 7.1.5 Households Households solve s.t. X max β t u(c t ) t=0 c t + a t+1 w t +(1+R t )a t As usual, the Euler condition gives And assuming CEIS, this reduces to u 0 (c t ) u 0 (c t+1 ) = β(1 + R t+1). c t+1 c t =[β(1 + R t+1 )] θ. 7.1.6 Resource Constraint Final goods are used either for consumption by households, or for production of intermediate goods in the intermediate sector, or for production of new blueprints in the innovation sector. The resource constraint of the economy is given by C t + K t + η N t = Y t, where C t = c t L, N t = N t+1 N t, and K t = R N t 0 X t,j dj. 135
George-Marios Angeletos 7.1.7 General Equilibrium Combining the formula for the value of innovation with the free-entry condition, we infer πl/r = V = η. It follows that the equilibrium interest rate is R = πl η = 1 α α A 1 2 1 α α 1 α L/η, which verifies our earlier claim that the interest rate is stationary. Theresourceconstraintreducesto C t Nt+1 + η 1 + X = Y t = AL 1 α X α, N t N t N t where X = xl = K t /N t. It follows that C t /N t is constant along the balanced growth path, and therefore C t,n t,k t, and Y t all grow at the same rate, γ. Combining the Euler condition with the equilibrium condition for the real interest rate, we conclude that the equilibrium growth rate is given by i 1+γ = β θ [1 + R] θ = β h1+ θ 1 α α A 1 2 θ 1 α α 1 α L/η Note that the equilibrium growth rate of the economy decreases with η, the cost of expanding product variety or producing new knowledge. The growth rate is also increasing in L or any other factor that increases the scale of the economy and thereby raises the profits of intermediate inputs and the demand for innovation. This is the (in)famous scale effect that is present in many models of endogenous technological change. Discuss... 136
Lecture Notes 7.1.8 Pareto Allocations and Policy Implications Consider now the problem of the social planner. Obviously, due to symmetry in production, the social planner will choose the same quantity of intermediate goods for all varieties: X t,j = X t = x t L for all j. Using this, we can write the problem of the social planner simply as maximizing utility, X max β t u(c t ), t=0 subject to the resource constraint C t + N t X t + η (N t+1 N t )=Y t = AL 1 α N t X α t, where C t = c t L. The FOC with respect to X t gives X t = x L, where x = A 1 1 α α 1 1 α represents the optimal level of production of intermediate inputs. The Euler condition, on the other hand, gives the optimal growth rate as i 1+γ = β θ [1 + R ] θ = β h1+ θ 1 α α A 1 1 θ 1 α α 1 α L/η, where R = 1 α α A 1 1 1 α α 1 α L/η represents that social return on savings. 137
George-Marios Angeletos Note that x = x α 1 1 α >x That is, the optimal level of production of intermediate goods is higher in the Pareto optimum than in the market equilibrium. This reflects simply the fact that, due to the monopolistic distortion, production of intermediate goods is inefficiently low in the market equilibrium. Naturally, the gap x /x is an increasing function of the mark-up 1/α. Similarly, R = R α 1 1 α >R. That is, the market return on savings (R) falls short of the social return on savings (R ), the gap again arising because of the monopolistic distortion in the intermediate good sector. It follows that 1+γ > 1+γ, so that the equilibrium growth rate is too low as compared to the Pareto optimal growth rate. Policy exercise: Consider three different types of government intervention: A subsidy on the production of intermediate inputs; an subsidy on the production of final goods (or the demand for intermediate inputs); and a subsidy on R&D. Which of these policies could achieve an increase in the market return and the equilibrium growth rate? Which of these policies could achieve an increases in the output of the intermediate good sector? Which one, or which combination of these policies can implement the first best allocations as a market equilibrium? 138
Lecture Notes 7.1.9 Introducing Skilled Labor and Human Capital topic covered in class notes to be completed 7.1.10 International Trade, Technology Diffusion, and other implications notes to be completed 7.2 Improving Product Quality: A Simple Model Before analyzing the full-fledge Aghion-Howitt model, we consider a simplified version that delivers most of the insights. The economy is populated by a large number of finitely-lived households. Eachproducerintheeconomyisan entrepreneur. Helives(bepresentinthemarket) for 1+T periods, where T is random. In particular, conditional on being alive in the present period, there is probability 1 n that the producer will be alive in the next period and a probability n that he will die (exit the market). n is constant over time and independent of the age of the producer. In each period, a mass n of existing producers dies, and a mass n of new producers is born. The population of producers is thus constant. In the first period of life, the producer is endowed with the aggregate level of knowledge in the economy. In the first period of life, he also has a fresh mind and may 139
George-Marios Angeletos engage in R&D activity. In the later periods of life, he is too old for coming up with good ideas and therefore engages only in production, not innovation. Young producers engage in R&D in order to increase the profits of their own productive activities later in life. But individual innovation has spillover effects to the whole economy. When a mass of producers generate new ideas, the aggregate level of knowledge in the economy increases proportionally with the production of new ideas. 7.2.1 R&D Technology Let V j t+1 denote the value of an innovation for individual j realized in period t and implemented in period t +1.Letz j t denote the amount of skilled labor that a potential innovator j employes in R&D and q(z j t ) the probability that such R&D activity will be successful. q : R [0, 1] represents the technology of producing innovations and it is assumed to be strictly increasing and strictly concave and satisfy the relevant Inada conditions: q(0) = 0, q 0 > 0 >q 00,q 0 (0) =, q 0 ( ) =0. The potential researcher maximizes q(z j t ) V j t+1 w t z j t. It follows that the optimal level of R&D is given by or q 0 (z j t )V j t+1 = w t z j t = g Ã! V j t+1 w t where the function g(v) (q 0 ) 1 (1/v) satisfies g(0) = 0, g 0 > 0, g( ) =. Note that the amount of labor devoted to R&D and the rate of innovation will be stationary only if both the value and the cost of innovation (V and w) grow at the same rate. 140
Lecture Notes 7.2.2 The Value of Innovation What determines the value of an innovation? For a start, let us assume a very simple structure. Let A j t represent the TFP of producer j in period t. The profits from his production are given by Π j t = A j t bπ where bπ represents normalized profits. We can endogenize π, but we won t do it here for simplicity. When a producer is born, he automatically learns what is the contemporaneous aggregate level of technology. That is, A j t = A t for any producer born in period t. In the first period of life, and only in that period, a producer has the option to engage in R&D. If his R&D activity fails to produce an innovation, the his TFP remains the same for the rest of his life. If instead his R&D activity is successful, then his TFP increases permanently by a factor 1+γ, for some γ>0. That is, for any producer j born in period t, and for all periods τ t +1in which he is alive, A j τ = A t if his R&D fails (1 + γ)a t if his R&D succeeds It follows that a successful innovation increases profits also by a factor 1+γ. That is, the innovation generates a stream of dividends equal to γa t bπ per period that the producer is alive. Since the producer expects to survive with a probability 1 n in each period, the expected present value of the increase in profits is given by V t+1 = X τ=t+1 µ τ 1 n (γa t bπ) =γbva t (7.1) 1+R 141